Uploaded March 2026 | Updated September 2026, 2 weeks ago
In this video, I show and describe at least 10 of my favorite Pi visualizations. Please let me know your favorite or what I should/could add!
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
If you want to see. more pi visualizations, check out my pi playlist:
youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq&si=VP6Ff6bpVfcp4T6d
If you want to know more about these visuals, here they are below:
1) Pi as almost 3 diameters - this is the definition and not based on any known published visual proof.
2) Pi epicycloid: this is also a classical notion and not documented.
3) Pi as a sum of arctangents in a rectangle is due to Rex Wu and comes from the March 2003 issue of The College Mathematics Journal (jstor.org/stable/3595783 ) see pages 115 and 138.
4) The pi from coprime integers is a classic: en.wikipedia.org/wiki/Coprime_integers
5) This visualization of peeling circumferences was inspired by several other similar visuals:
from @MinutePhysics : youtu.be/whYqhpc6S6g
For more information about this construction, see
personal.math.ubc.ca/~cass/courses/m446-03/exhaustion.pdf
or check out this nice survey article by David Richeson from the May 2015 issue of The College Math Journal: doi.org/10.4169/college.math.j.46.3.162 .
6) The area via wedges proof can be traced to both Satō Moshun and Leonardo da Vinci (see Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, archive.org/details/historyofjapanes00smituoft, page 130-132 and Beckmann, Petr (1976), A History of Pi, St. Martin's Griffin, page 19).
You can also read more about this in a great NYT article by Steven Strogatz: archive.nytimes.com/opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit
7) The Madhava series visualization was inspired by an article from Mitsuo Kobayashi that appeared in the April 2014 issue of Mathematics Magazine (jstor.org/stable/10.4169/math.mag.87.2.145
) see pages 145-150. That article features this dissection proof attributed to Viggo Brun.
8) The pi in a 3-4-5 triangle visual is based on a visual proof from Proofs without Words II by Roger B. Nelsen (MAA, 2000) (bookstore.ams.org/clrm-14/) page 14
9) This false argument was suggested to me by Jeff Stuart as a nice alternative to the more classic Pi = 4 argument, which can be found many places.
10) The pi as simulated by dartboards is also a classic and has been done many times in many videos.
#manim #irrational #Pi #mathvideo #math #mtbos #animation #iteachmath #mathematics #piday #shorts #trigonometry #tangent #triangle #rectangle #identities #infiniteseries #series #dissectionproof #dissection #leibniz #leibnizformula #infinitesum #calculus #integral #proofwithoutwords #visualproof
To learn more about animating with manim, check out:
https://manim.community
In this video, I show and describe at least 10 of my favorite Pi visualizations. Please let me know your favorite or what I should/could add!
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
If you want to see. more pi visualizations, check out my pi playlist:
youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq&si=VP6Ff6bpVfcp4T6d
If you want to know more about these visuals, here they are below:
1) Pi as almost 3 diameters - this is the definition and not based on any known published visual proof.
2) Pi epicycloid: this is also a classical notion and not documented.
3) Pi as a sum of arctangents in a rectangle is due to Rex Wu and comes from the March 2003 issue of The College Mathematics Journal (jstor.org/stable/3595783 ) see pages 115 and 138.
4) The pi from coprime integers is a classic: en.wikipedia.org/wiki/Coprime_integers
5) This visualization of peeling circumferences was inspired by several other similar visuals:
from @MinutePhysics : youtu.be/whYqhpc6S6g
For more information about this construction, see
personal.math.ubc.ca/~cass/courses/m446-03/exhaustion.pdf
or check out this nice survey article by David Richeson from the May 2015 issue of The College Math Journal: doi.org/10.4169/college.math.j.46.3.162 .
6) The area via wedges proof can be traced to both Satō Moshun and Leonardo da Vinci (see Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, archive.org/details/historyofjapanes00smituoft, page 130-132 and Beckmann, Petr (1976), A History of Pi, St. Martin's Griffin, page 19).
You can also read more about this in a great NYT article by Steven Strogatz: archive.nytimes.com/opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit
7) The Madhava series visualization was inspired by an article from Mitsuo Kobayashi that appeared in the April 2014 issue of Mathematics Magazine (jstor.org/stable/10.4169/math.mag.87.2.145
) see pages 145-150. That article features this dissection proof attributed to Viggo Brun.
8) The pi in a 3-4-5 triangle visual is based on a visual proof from Proofs without Words II by Roger B. Nelsen (MAA, 2000) (bookstore.ams.org/clrm-14/) page 14
9) This false argument was suggested to me by Jeff Stuart as a nice alternative to the more classic Pi = 4 argument, which can be found many places.
10) The pi as simulated by dartboards is also a classic and has been done many times in many videos.
#manim #irrational #Pi #mathvideo #math #mtbos #animation #iteachmath #mathematics #piday #shorts #trigonometry #tangent #triangle #rectangle #identities #infiniteseries #series #dissectionproof #dissection #leibniz #leibnizformula #infinitesum #calculus #integral #proofwithoutwords #visualproof
To learn more about animating with manim, check out:
https://manim.community
